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\hypersetup{
  pdftitle={Ch4 生命年金},
  pdfauthor={滕帆},
  hidelinks,
  pdfcreator={LaTeX via pandoc}}

\title{Ch4 生命年金}
\subtitle{保险精算学}
\author{滕帆}
\date{2023-09}
\institute{浙大宁波理工学院金融学系}

\begin{document}
\frame{\titlepage}

\begin{frame}[allowframebreaks]
  \tableofcontents[hideallsubsections]
\end{frame}
\hypertarget{ux77e5ux8bc6ux56deux987e}{%
\section{知识回顾}\label{ux77e5ux8bc6ux56deux987e}}

\begin{frame}{知识回顾}
\begin{itemize}
\tightlist
\item
  定期寿险给付保险金现值
\end{itemize}

\[
Z=
\begin{cases}
v^T& 0\leq T \leq n \\
0& T>n 
\end{cases}
\]

\begin{itemize}
\item
  身故年末给付的定期寿险精算现值 \[
  \Ax{\termxn} = \qx{x} v+\qx[1|]{x} v^2+ \ldots + \qx[n-1|]{x} v^n
  \]
\item
  即死即付的定期寿险精算现值
\end{itemize}

\[
\Ax*{\termxn}=\int_0^n{v^t \px[t]{x}\mu_{x+t}dt}
\]
\end{frame}

\begin{frame}
\textbf{终身寿险精算现值}

\begin{itemize}
\item
  身故年末给付的终身寿险精算现值 \[
  \Ax{x} = \qx{x} v+\qx[1|]{x} v^2+ \ldots + \qx[\omega-1|]{x} v^{\omega}
  \]
\item
  即死即付的终身寿险精算现值
\end{itemize}

\[
\Ax*{x}=\int_0^\omega{v^t \px[t]{x}\mu_{x+t}dt}
\]
\end{frame}

\hypertarget{ux751fux547dux5e74ux91d1ux5206ux7c7b}{%
\section{生命年金分类}\label{ux751fux547dux5e74ux91d1ux5206ux7c7b}}

\begin{frame}{生命年金分类}
\textbf{年金}：被保险人生存时得到的一系列给付。与人寿保险相对应，可以称为``生存年金''、``生命年金''。

\begin{itemize}
\tightlist
\item
  按保险期限：

  \begin{itemize}
  \tightlist
  \item
    终身年金\(\ax{x}\)、\(\ax**{x}\)、\(\ax*{x}\)、\(\ax{x}[(m)]\)
  \item
    定期年金\(\ax{\endowxn}\)
  \item
    延期年金\(\ax[s|]{x}\)
  \end{itemize}
\item
  年金的其他特殊形式：

  \begin{itemize}
  \tightlist
  \item
    定期确定年金：确定给付年数或保证最低给付年数
  \item
    指数化年金
  \item
    联合生存年金
  \end{itemize}
\end{itemize}
\end{frame}

\hypertarget{ux751fux5b58ux5e74ux521dux7ed9ux4ed8ux7ec8ux8eabux5e74ux91d1}{%
\section{生存年初给付终身年金}\label{ux751fux5b58ux5e74ux521dux7ed9ux4ed8ux7ec8ux8eabux5e74ux91d1}}

\begin{frame}{定义}
\protect\hypertarget{ux5b9aux4e49}{}
\begin{itemize}
\item
  \textbf{年金}：被保险人生存时得到的一系列给付。
\item
  \textbf{终身}年金：保险期限直至被保险人身故的年金。
\item
  \textbf{生存年初给付}终身年金：保险金在生存年初给付的终身年金。
\end{itemize}
\end{frame}

\begin{frame}{精算现值计算}
\protect\hypertarget{ux7cbeux7b97ux73b0ux503cux8ba1ux7b97}{}
Step1: 考虑期初终身年金，列出现金流，并计算现值Y

\[
\begin{aligned}
Y&=1+v+v^2+\ldots+v^K\\
&=\ax**{\angl{K+1}}
\end{aligned}
\]

Step2: 确定Y的概率分布

\[
\begin{aligned}
Pr(Y=\ax**{\angl{K+1}})
&=Pr(K=k)\\
&=\px[k]{x}\qx{x+k}
\end{aligned}
\]
\end{frame}

\begin{frame}
Step3: 计算终身年金给付现值Y的期望（精算现值）

\[
\ax**{x}=\sum_{k=0}^{\infty}\ax**{\angl{k+1}}\px[k]{x}\qx{x+k}
\]

step4: 计算终身年金给付现值Y的方差

\[
Var(Y)=\frac{Var(Z)}{d^2}
\]
\end{frame}

\begin{frame}
\textbf{生存年初给付终生年金}与\textbf{身故年末给付终身寿险}之间的关系

\begin{itemize}
\item
  \(Z\)为身故年末给付终身寿险的给付现值，其精算现值为\(\Ax{x}=E(Z)\)
\item
  \(Y\)为生存年初给付终身年金的给付现值，其精算现值为\(\ax**{x}=E(Y)\)
\end{itemize}

结论1：\(Y=\frac{1-Z}{d}\)

结论2：\(\ax**{x}=\frac{1-\Ax{x}}{d}\)

结论3：\(d\ax**{x}+\Ax{x}=1\)

结论4：\(Var(Y)=\frac{Var(Z)}{d^2}=\frac{\Ax[][2]{x}-(\Ax{x})^2}{d^2}\)
\end{frame}

\hypertarget{ux751fux5b58ux5e74ux521dux7ed9ux4ed8ux5b9aux671fux5e74ux91d1}{%
\section{生存年初给付定期年金}\label{ux751fux5b58ux5e74ux521dux7ed9ux4ed8ux5b9aux671fux5e74ux91d1}}

\begin{frame}{生存年初给付定期年金}
Step1: n年期期初生命年金的现值变量Y为 \[
Y=
\begin{cases}
\ax**{\angl{K+1}}& K=0,1,2, \ldots ,n-1 \\
\ax**{\angln}& K\geq n 
\end{cases}
\] Step2: 求Y的期望（精算现值）

\[
\begin{aligned}
\ax**{\endowxn}&=\sum_{k=0}^{n-1}\ax**{\angl{k+1}}\px[k]{x}\qx{x+k}+\ax**{\angln}\px[n]{x} \\
\ax**{\endowxn}&=\sum_{k=0}^{n-1}v^k\px[k]{x}
\end{aligned}
\]
\end{frame}

\begin{frame}
\textbf{生存年初给付定期年金}与\textbf{身故年末给付定期寿险}之间关系

结论与终身年金相似：

结论1：\(Y=\frac{1-Z}{d}\)

结论2：\(\ax**{\endowxn}=\frac{1-\Ax{\endowxn}}{d}\)

结论3：\(d\ax**{\endowxn}+\Ax{\endowxn}=1\)
\end{frame}

\hypertarget{ux5e74ux5185ux7ed9ux4ed8ux591aux6b21mux6b21ux7684ux751fux547dux5e74ux91d1}{%
\section{年内给付多次(m次)的生命年金}\label{ux5e74ux5185ux7ed9ux4ed8ux591aux6b21mux6b21ux7684ux751fux547dux5e74ux91d1}}

\begin{frame}{年内给付多次(m次)的生命年金}
定义：一年给付m次，每次期初给付\(1/m\)的生命年金。该终身生命年金的精算现值记作

\[
\ax**{x}[(m)]
\]

不加证明地给出计算公式为

\begin{itemize}
\item
  \(d^{(m)}\ax**{x}[(m)]+\Ax{x}[(m)]=1\)
\item
  \(\ax**{x}[(m)]=\frac{1}{d^{(m)}}-\frac{1}{d^{(m)}}\Ax{x}[(m)]\)
\end{itemize}
\end{frame}

\begin{frame}
当\(m\to \infty\)时，即时时给付终身年金的精算现值

\begin{itemize}
\item
  \(\delta \ax*{x} +\Ax*{x}=1\)
\item
  \(\ax*{x}=\frac{1-\Ax*{x}}{\delta}\)
\end{itemize}
\end{frame}

\hypertarget{ux751fux5b58ux671fux521dux7ed9ux4ed8ux5ef6ux671fsux5e74ux7ec8ux8eabux5e74ux91d1}{%
\section{\texorpdfstring{生存期初给付延期\(s\)年终身年金}{生存期初给付延期s年终身年金}}\label{ux751fux5b58ux671fux521dux7ed9ux4ed8ux5ef6ux671fsux5e74ux7ec8ux8eabux5e74ux91d1}}

\begin{frame}{生存期初给付延期\(s\)年终身年金}
定义：0时刻投保，但在生存至\(s\)年后每年初给付1元，直至身故。

令Y为该年金现值，则

\[
Y=
\begin{cases}
0& K=0,1,2,\ldots,s-1 \\
v^s+v^{s+1}+v^{s+2}+\ldots+v^K& K=s,s+1,s+2,\ldots 
\end{cases}
\]
\end{frame}

\begin{frame}
该年金的精算现值为

\[
\ax**[s|]{x}=\px[s]{x}v^m\ax**{x}
\] 或者

\[
\ax**[s|]{x}=\ax**{x}-\ax**{x:\angl{s}}
\]
\end{frame}

\hypertarget{ux751fux547dux5e74ux91d1ux7cbeux7b97ux73b0ux503cux8ba1ux7b97ux53caux5176rux5b9eux73b0}{%
\section{生命年金精算现值计算及其R实现}\label{ux751fux547dux5e74ux91d1ux7cbeux7b97ux73b0ux503cux8ba1ux7b97ux53caux5176rux5b9eux73b0}}

\begin{frame}[fragile]{\texttt{lifecontingencies}中的\texttt{axn}函数简介}
\protect\hypertarget{lifecontingenciesux4e2dux7684axnux51fdux6570ux7b80ux4ecb}{}
\begin{verbatim}
axn(actuarialtable,             #精算表
    x,                          #投保年龄
    n,                          #保险期限
    i = actuarialtable@interest,#预定利率 
    m,                          #延期时间
    k = 1,                      #一年内多次给付
    type = "EV",                #计算类型，EV为精算现值
    power=1,                    #指数
    payment = "advance",        #何时给付：
                                   #advance或due为期初付
                                   #immediate或arrears为期末付
    ...                         #计算pxt的其他参数
    )
\end{verbatim}
\end{frame}

\begin{frame}[fragile]{Step1: 环境准备}
\protect\hypertarget{step1-ux73afux5883ux51c6ux5907}{}
\begin{Shaded}
\begin{Highlighting}[]
\DocumentationTok{\#\#1.清空内存}
\FunctionTok{rm}\NormalTok{(}\AttributeTok{list=}\FunctionTok{ls}\NormalTok{())}
\FunctionTok{gc}\NormalTok{()}

\DocumentationTok{\#\#2.加载必要的R包}
\CommentTok{\# library(knitr)}
\CommentTok{\# library(tidyverse)}
\FunctionTok{library}\NormalTok{(lifecontingencies)}
\CommentTok{\# library(actuar)}
\end{Highlighting}
\end{Shaded}
\end{frame}

\begin{frame}[fragile]{Step2: 生成精算表}
\protect\hypertarget{step2-ux751fux6210ux7cbeux7b97ux8868}{}
\begin{Shaded}
\begin{Highlighting}[]
\DocumentationTok{\#\#3.生成精算表}
\DocumentationTok{\#\#\#3.1导入生命表}
\NormalTok{clt}\OtherTok{\textless{}{-}}\FunctionTok{read.csv}\NormalTok{(}\StringTok{"../lifetable/lifetable\_cn.csv"}\NormalTok{)}
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             \AttributeTok{x=}\NormalTok{clt}\SpecialCharTok{@}\NormalTok{x,}
             \AttributeTok{lx=}\NormalTok{clt}\SpecialCharTok{@}\NormalTok{lx,}\AttributeTok{interest=}\FloatTok{0.03}\NormalTok{)}
\end{Highlighting}
\end{Shaded}
\end{frame}

\begin{frame}[fragile]{Step3：计算年金现值}
\protect\hypertarget{step3ux8ba1ux7b97ux5e74ux91d1ux73b0ux503c}{}
\begin{Shaded}
\begin{Highlighting}[]
\DocumentationTok{\#\#4.计算年金现值}
\DocumentationTok{\#\#\#4.1每年给付的终身年金}
\FunctionTok{axn}\NormalTok{(clt.act,}\AttributeTok{x=}\DecValTok{22}\NormalTok{,}\AttributeTok{payment =} \StringTok{"due"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

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\end{Shaded}

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\end{verbatim}
\end{frame}

\begin{frame}[fragile]
\begin{Shaded}
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\DocumentationTok{\#\#\#4.2每年m次给付的终身年金}
\FunctionTok{axn}\NormalTok{(clt.act,}\AttributeTok{x=}\DecValTok{22}\NormalTok{,}\AttributeTok{k=}\DecValTok{1}\NormalTok{,}\AttributeTok{payment =} \StringTok{"due"}\NormalTok{)}
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\begin{frame}[fragile]
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\DocumentationTok{\#\#\#4.3定期年金}
\FunctionTok{axn}\NormalTok{(clt.act,}\AttributeTok{x=}\DecValTok{22}\NormalTok{,}\AttributeTok{n=}\DecValTok{38}\NormalTok{,}\AttributeTok{payment =} \StringTok{"due"}\NormalTok{)}
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\begin{frame}[fragile]
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\begin{frame}
Any Questions?
\end{frame}

\end{document}
